On the competitive exclusion principle for continuously distributed populations
Résumé
We investigate the competitive exclusion principle in the case of a continuously distributed initial population. We introduce an ordinary differential equations model of species competing for a single resource, whose efficiency is encoded by a continuous variable (the "trait variable") living in a Euclidean space. The differential equation is solved in a measure space to allow the observation of the natural concentration of the species distribution on specific traits. We show the concentration of the distribution on the maximizing set of the fitness function in the sense of the Kantorovitch-Rubinstein metric. In the case when the initial weight of the maximizing set is positive, we give a precise description of the convergence of the orbit, including a formula for the asymptotic distribution. We also investigate precisely the case of a finite number of regular global maxima and show that the initial distribution may have an influence on the support of the eventual distribution. In particular, the natural process of competition is not always selecting a unique species, as it would be predicted by the competitive exclusion principle, but several species may coexist as long as they maximize the fitness function. In many cases it is possible to compute the eventual distribution of the surviving competitors. In some configurations, species that maximize the fitness may still get extinct depending on the shape of the initial distribution and on an other parameter of the model, and we provide a way to characterize when this unexpected extinction happens.
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